Methods for generating variates from probability distributions

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Diverse probabilistic results are used in the design of random univariate generators. General methods based on these are classified and relevant theoretical properties derived. This is followed by a comparative review of specific algorithms currently available for continuous and discrete univariate distributions. A need for a Zeta generator is established, and two new methods, based on inversion and rejection with a truncated Pareto envelope respectively are developed and compared. The paucity of algorithms for multivariate generation motivates a classification of general methods, and in particular, a new method involving envelope rejection with a novel target distribution is proposed. A new method for generating first passage times in a Wiener Process is constructed. This is based on the ratio of two random numbers, and its performance is compared to an existing method for generating inverse Gaussian variates. New "hybrid" algorithms for Poisson and Negative Binomial distributions are constructed, using an Alias implementation, together with a Geometric tail procedure. These are shown to be robust, exact and fast for a wide range of parameter values. Significant modifications are made to Atkinson's Poisson generator (PA), and the resulting algorithm shown to be complementary to the hybrid method. A new method for Von Mises generation via a comparison of random numbers follows, and its performance compared to that of Best and Fisher's Wrapped Cauchy rejection method. Finally new methods are proposed for sampling from distribution tails, using optimally designed Exponential envelopes. Timings are given for Gamma and Normal tails, and in the latter case the performance is shown to be significantly better than Marsaglia's tail generation procedure.Governors of Dundee College of Technolog

Random Texts Do Not Exhibit the Real Zipf's Law-Like Rank Distribution

Zipf's law states that the relationship between the frequency of a word in a text and its rank (the most frequent word has rank , the 2nd most frequent word has rank ,...) is approximately linear when plotted on a double logarithmic scale. It has been argued that the law is not a relevant or useful property of language because simple random texts - constructed by concatenating random characters including blanks behaving as word delimiters - exhibit a Zipf's law-like word rank distribution.In this article, we examine the flaws of such putative good fits of random texts. We demonstrate - by means of three different statistical tests - that ranks derived from random texts and ranks derived from real texts are statistically inconsistent with the parameters employed to argue for such a good fit, even when the parameters are inferred from the target real text. Our findings are valid for both the simplest random texts composed of equally likely characters as well as more elaborate and realistic versions where character probabilities are borrowed from a real text.The good fit of random texts to real Zipf's law-like rank distributions has not yet been established. Therefore, we suggest that Zipf's law might in fact be a fundamental law in natural languages

Bayesian inference for transportation origin-destination matrices: the Poisson-inverse Gaussian and other Poisson mixtures

Transportation origin–destination analysis is investigated through the use of Poisson mixtures by introducing covariate‐based models which incorporate different transport modelling phases and also allow for direct probabilistic inference on link traffic based on Bayesian predictions. Emphasis is placed on the Poisson–inverse Gaussian model as an alternative to the commonly used Poisson–gamma and Poisson–log‐normal models. We present a first full Bayesian formulation and demonstrate that the Poisson–inverse Gaussian model is particularly suited for origin–destination analysis because of its desirable marginal and hierarchical properties. In addition, the integrated nested Laplace approximation is considered as an alternative to Markov chain Monte Carlo sampling and the two methodologies are compared under specific modelling assumptions. The case‐study is based on 2001 Belgian census data and focuses on a large, sparsely distributed origin–destination matrix containing trip information for 308 Flemish municipalities

Estimation and Applications of Quantile Regression for Binary Longitudinal Data

This paper develops a framework for quantile regression in binary longitudinal data settings. A novel Markov chain Monte Carlo (MCMC) method is designed to fit the model and its computational efficiency is demonstrated in a simulation study. The proposed approach is flexible in that it can account for common and individual-specific parameters, as well as multivariate heterogeneity associated with several covariates. The methodology is applied to study female labor force participation and home ownership in the United States. The results offer new insights at the various quantiles, which are of interest to policymakers and researchers alike

Aspectos computacionales en la estimación de incertidumbres de ensayo por el Método de Monte Carlo = Computational aspects in uncertainty estimation by Monte Carlo Method

El propósito de este trabajo es analizar los distintos aspectos relacionados al desarrollo de una aplicación informáticapara la estimación de incertidumbres de ensayo por el método de Monte Carlo, independiente de plataformas de cálculo como MS Excel, MathLab o R. Se analizan las dificultades y posibles soluciones en cada una de las etapas necesarias para alcanzar este objetivo, el algoritmo para la creación de un intérprete de ecuaciones, la generación de números pseudo-aleatorios con las distribuciones de probabilidad más frecuentes y el tratamiento de incertidumbres Tipo A por este método. Finalmente se hace un estudio comparativo de los resultados obtenidos con la aplicación generada, el método clásico (GUM) y la misma simulación realizada con el Software R. Este estudio se realiza sobre el cálculo de la densidad del aire según ecuación CIPM, la presión generada por una balanza de presión y la estandarización de una solución de hidróxido de sodio de acuerdo al ejemplo A2 de la guía EURACHEM / CITEC CG 4

Aplicação e validação de um simulador estocástico de variáveis climáticas. O caso da precipitação

A carência de informação meteorológica é uma realidade comum a todas as regiões de Portugal. Muitos são os estudos que estão condicionados a esta lacuna o que conduz, mormente, à realização de estudos e projectos revestidos de incertezas e, talvez, à inibição de certas intervenções no âmbito da Hidrologia e Hidráulica. Neste trabalho descrevem-se a metodologia e os dados necessários para o ajustamento e aplicação do gerador climático CLIGEN no Sul de Portugal (VALE FORMOSO). Avalia-se a performance do CLIGEN na simulação da precipitação diária e mensal. A importância do conceito da normal climatológica é outro aspecto evidenciado no âmbito deste trabalho. Apresentase a análise de sensibilidade do modelo aos diferentes parâmetros de entrada – considerando três períodos distintos de registos de dados climáticos (10, 20 e 30 anos). O CLIGEN gera parâmetros climáticos indispensáveis para a aplicação de vários modelos hidrossedimentológicos dentre os quais se destacam o WEPP (Water Erosion Prediction Project), EPIC (Erosion/ Produtivity Impact Calculator), SWRRB (Simulator for Water Resources in Rural Basins), AGNPS (Agricultural Nonpoint Source Pollution Model) e CREAMS (Chemicals, Runoff, and Erosion from Agricultural Management System). Este estudo permitiu concluir que o modelo reproduz de modo significativo o padrão da precipitação e também evidencia que a extensão da série de precipitação considerada exerce considerável influência nos resultados

Factors affecting bird counts and their influence on density estimates

Variable area surveys are used in large geographic regions to estimate the density of birds distributed over a region. If some birds go undetected, a measure of the effective area surveyed, the amount of area occupied by the birds detected, is needed. The effective area surveyed is determined by observational, biological, and environmental factors relating to detectability. It has been suggested that density estimates are inaccurate, and that it is risky to compare bird populations intraspecifically over time and space, since factors influencing bird counts will vary. There have been several controversial studies where variable area survey density estimates were evaluated using density estimates calculated from spot mapping as the standard for comparison. Spot mapping itself is an unproven estimator that the previously mentioned factors also influence. Without a known population density, determining how the different density estimators perform is difficult to access. Variable area surveys of inanimate objects whose densities were known have been conducted under controlled circumstances with results generally supporting the variable area survey method, but time and inability to control for all factors limit the application of this type of study. A simulation program that distributes over a region vegetation and a known density of birds, and then simulates the process of gathering bird detection data is one tool accessible to evaluate variable area density estimates. Within such a simulation study various observational, biological, and environment factors could be introduced. This thesis introduces such a simulation program, VABS, that was written with the objectives of identifying factors that influence bird counts and determining the limitations of the variable area survey. Within this thesis are discussions concerning the several factors that have been identified as influencing bird counts and the effects that these factors had on the Fourier series, exponential power series, and Cum-D density estimates when these factors were simulated in VABS. Critical assumptions of the variable area survey are identified, and the ability of the variable area survey to estimate density for different detectability curve is examined. Also included are discussions on the topics of pooling data gathered under different detectabilities and monitoring population trends

Aplicação e validação de um simulador estocástico de variáveis climáticas. O caso da precipitação

[PT] A carência de informação meteorológica é uma realidade comum a todas as regiões de Portugal. Muitos são os estudos que estão condicionados a esta lacuna o que conduz, mormente, à realização de estudos e projectos revestidos de incertezas e, talvez, à inibição de certas intervenções no âmbito da Hidrologia e Hidráulica. Neste trabalho descrevem-se a metodologia e os dados necessários para o ajustamento e aplicação do gerador climático CLIGEN no Sul de Portugal (VALE FORMOSO). Avalia-se a performance do CLIGEN na simulação da precipitação diária e mensal. A importância do conceito da normal climatológica é outro aspecto evidenciado no âmbito deste trabalho. Apresentase a análise de sensibilidade do modelo aos diferentes parâmetros de entrada – considerando três períodos distintos de registos de dados climáticos (10, 20 e 30 anos). O CLIGEN gera parâmetros climáticos indispensáveis para a aplicação de vários modelos hidrossedimentológicos dentre os quais se destacam o WEPP (Water Erosion Prediction Project), EPIC (Erosion/ Produtivity Impact Calculator), SWRRB (Simulator for Water Resources in Rural Basins), AGNPS (Agricultural Nonpoint Source Pollution Model) e CREAMS (Chemicals, Runoff, and Erosion from Agricultural Management System). Este estudo permitiu concluir que o modelo reproduz de modo significativo o padrão da precipitação e também evidencia que a extensão da série de precipitação considerada exerce considerável influência nos resultados.Fernandes Lima, HM.; Pereida Da Mata, I.; Fernandes Lima, AV. (2005). Aplicação e validação de um simulador estocástico de variáveis climáticas. O caso da precipitação. Ingeniería del agua. 12(1):27-37. https://doi.org/10.4995/ia.2005.2549OJS2737121Arnold, J.G. & Williams, J.R. (1989). Stochastic Generation of Internal Storm Structure. Trans. 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Publicação nº 4/94, CEHIDRO - Centro de Estudos de Hidrossistemas, Instituto Superior Técnico, Lisboa.Viessman, W.Jr.; Knapp, J.W.; Lewis, G.L.; & Harbough, T.E. (1977). Introduction to Hydrology, Dun - Donnelly, New York.World Meteorological Organization, WMO (1989). Calculation of Monthly and Annual 30-Year Standard Normals, WCDP-Nº. 10, WMO-TD/Nº 341, Geneva: World Meteorological Organization.World Meteorological Organization, WMO (1995). Climatic Change, Technical Note Nº 79, WMO - Nº 195.TP100, 79 p